Problem C. 1264. (December 2014)

C. 1264. The interior angle bisector drawn from vertex \(\displaystyle A\) of triangle \(\displaystyle ABC\) intersects the opposite side at point \(\displaystyle P\), and the perpendicular bisector of line segment \(\displaystyle AP\) intersects side \(\displaystyle AC\) at point \(\displaystyle Q\). Given the line segments \(\displaystyle AB\), \(\displaystyle AQ\), and the angle \(\displaystyle BAQ\), express the area of quadrilateral \(\displaystyle ABPQ\).